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Conchoid of Nicomedes (conchoid of a straight line) for various values of the set distance. The black line is the reference curve (C in the definition), and the red dot is the fixed point (P).
A conchoid of circle
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From Latin concha (mussel) (from Ancient Greek κόγχη (kónkhē)) +‎ -oid, referring to the curved outline of a mussel shell.


conchoid (plural conchoids)

  1. (mathematics, geometry) Any of a family of curves defined as the locus of points p, such that each p is on a line that passes through a given fixed point P and intersects a given curve, C, and the distance from p to the point of intersection with C is a specified constant (note that for nontrivial cases two such points p satisfy the criteria, and the resultant curve has two parts).
    The conchoid of a circle with respect to a point on the circle is a cardioid if the fixed distance is equal to the diameter of the circle.
    The Conchoid of Nicomedes is the conchoid of a straight line with respect to a point not on the line.
    • 1815, Charles Hutton, Pappus, entry in A Philosophical and Mathematical Dictionary, Volume 2, page 147,
      He next treats of the properties of the Conchoid, which Nicomedes invented for doubling the cube; applying it to the solution of certain problems concerning Inclinations, with the finding of two mean proportionals, and cubes in any proportion whatever.
    • 1982, J. Lee Kavanau, Curves and Symmetry[1], volume 1, page 3:
      The classical conchoid construction is a non-orthogonal polar-curvilinear construction in which equal distances along a line are marked off from its point of intersection with a curve for various positions of the line as it rotates about a point.
    • 2007, James Stewart, Single Variable Calculus[2], volume 2, page 662:
      These curves are called conchoids of Nicomedes after the ancient Greek scholar Nicomedes. He called them conchoids because the shape of their outer branches resembles that of a conch shell or mussel shell.
    • 2009, Niccolò Guicciardini, Isaac Newton on Mathematical Certainty and Method[3], page 68:
      One of the best choices is the conchoid, according to Newton the simplest curve after the circle.
  2. (geology) A conchoidal fracture in rock.
    • 1948, Tennessee Valley Authority, The Hiwassee Valley Projects, Technical Report, Issue 5, Volume 2, page 359,
      Conchoids of sound rock, from a few feet to 20 or more feet in diameter, entirely surrounded by comparatively thin layers of weathered material, were frequently encountered, sometimes in adjacent series.

Usage notes[edit]

The fixed point (P) of the construction may be referred to as the focus of the conchoid; it may also be defined as the origin (of a Cartesian coordinate system) or the pole (if polar coordinates are used), and potentially referred to accordingly. The curve C is an example of a directrix.

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